One possible way of proving the famous and important Riemann hypothesis would be to realize the Riemann zeta zeroes as the eigenvalues of some self-adjoint operator, using self-adjointness to show that the non-trivial zeroes all lie on the critical line. This is known as the Hilbert-Polya conjecture. Interpreting such a self-adjoint operator as an observable of some quantum system, one can try to describe the Riemann zeta zeroes in physical terms, making the Hilbert-Polya conjecture an interesting border case between mathematics and physics.

Major work on the Hilbert-Polya conjecture was done by Alain Connes. In his framework, the prolate spheroidal differential operator plays an important but auxiliary role. Before Connes’ work, this second-order differential operator was mainly known for its use in signal analysis, specifically in Slepian’s work on signals which are both time-limited and band-limited. In a more recent article, Connes and Moscovici show that the relation between the prolate spheroidal operator and the Riemann zeta function might be deeper than it seems. They show that a certain self-adjoint realization of the prolate spheroidal operator on the entire real line has discrete spectrum that is asymptotically similar to the squares of the Riemann zeta zeroes. This surprising discovery allows them to construct an operator which approximately solves the Hilbert-Polya conjecture.

Though Connes and Moscovici’s article is wonderful and inspiring, it is tersely written. Hence, in this bachelor thesis, details have been provided to the reasoning of the article. In addition, the physical interpretations of the Riemann zeta zeroes, which support the Hilbert-Polya conjecture, have also been presented.

In this report a semi-analytic solution to the Laplacian magic window is proposed. The Laplacian magic window is a term recently introduced in 2017[2]. When a uniform wavefront hits a refractive surface, it creates an illumination distribution behind the surface. When the curvature of the surface is sufficiently small, it can be related linearly to the target illumination, and thus creates a ‘magic window’. The main idea of the semi-analytic solution is that a target illumination or a surface given in terms of Zernike polynomials can be solved analytically and expressed again in Zernike polynomials. The Zernike polynomials are set of complete and orthogonal polynomials that are already used in the field of optics to describe wavefronts of optical surfaces. The results of the semi-analytic solution agree qualitatively with the numeric results and work for complex and diverse inputs. The implementation is done in 2D, but the method is general enough so it can be extended to 3D.